Worksheet: Applications on Sine and Cosine Laws

In this worksheet, we will practice using the cosine and sine law to solve real-world problems.

Q1:

𝐴𝐵𝐶 is a triangle, where 𝐷 is the midpoint of
𝐵𝐶, 𝑚∠𝐵=28∘,
𝑚∠𝐴=69∘, and
𝑎=20cm. Find the length of 𝐴𝐷 giving the answer to two decimal places.

Q2:

𝐴𝐵𝐶 is a triangle, where 𝑎=19 cm, 𝑏=9 cm, and 𝑚∠𝐶=45∘. Find the radius of the circumcircle giving the answer to two decimal places.

Q3:

𝐴𝐵𝐶 is a triangle where 𝑎=26 cm, 𝑏=22 cm
and 𝑐=6 cm.
Find the radius of the circumcircle giving the answer to three decimal places.

Q4:

𝐴𝐵𝐶 is a triangle where 𝑚∠𝐴=30∘, the ratio between 𝑏 and 𝑐 is √3∶2, and the area of the circumcircle is 225𝜋 cm2. Find the perimeter of triangle 𝐴𝐵𝐶 giving the answer to the nearest centimeter.

Q5:

𝐴𝐵𝐶𝐷 is a quadrilateral where 𝐴𝐵=16cm, 𝑚∠𝐴𝐷𝐵=40∘,
𝑚∠𝐷𝐵𝐴=100∘, 𝐵𝐶=21cm and 𝐷𝐶=9cm.
Find 𝑚∠𝐵𝐶𝐷 giving the answer to the nearest second and the area of 𝐵𝐶𝐷 giving the answer to three decimal places.

A11110′6′′∘, 88.123 cm2

B2333′23′′∘, 37.767 cm2

C6923′58′′∘, 134.283 cm2

D4516′30′′∘, 67.142 cm2

Q6:

𝐴𝐵𝐶𝐷 is a quadrilateral where
𝑚∠𝐴𝐵𝐶=90∘, 𝑚∠𝐵𝐴𝐷=41∘, 𝐴𝐵=𝐴𝐷=30.9cm
and 𝐵𝐷=𝐵𝐶. Find the area of
𝐴𝐵𝐶𝐷 giving the answer to two decimal places.

Q7:

Find the area of 𝐴𝐵𝐶𝐷 given 𝐸 is
the point of intersection of
𝐴𝐶 and 𝐵𝐷, 𝐴𝐸=5cm,
𝐸𝐶=8.9cm,
𝐸𝐷=7.7cm,
and 𝑚∠𝐴𝐸𝐵=80∘.
Give the answer to the nearest square centimeter.

Q8:

𝐴𝐵𝐶𝐷 is a parallelogram where 𝑀
is the point of intersection to the
diagonals, 𝐴𝐶=21.1cm, 𝑚∠𝐴𝑀𝐷=8054′∘
and 𝑚∠𝐶𝐴𝐵=5354′∘.
Find the area of the parallelogram giving the answer to two decimal places.

Q9:

In a parallelogram 𝐴𝐵𝐶𝐷,
𝐵𝐶=8inches
and 𝐴𝐵=5inches,
where 𝑚∠𝐴𝐵𝐶=134∘. Work out the length
of 𝐴𝐶. Give your answer to 2 decimal places.

Q10:

The side length of a regular pentagon 𝐴𝐵𝐶𝐷𝐸 is 25.81 cm. Find the length of the diagonal 𝐴𝐶 giving the answer to two decimal places.

Q11:

In a triangle 𝐴𝐵𝐶, 𝐴𝐵=3,
𝐴𝐶=8, and
𝑚∠𝐵𝐴𝐶=52∘.

If 𝐵𝐷 is the altitude from 𝐵, what is 𝐵𝐷 to 2 decimal places?

Calculate the area of the triangle 𝐴𝐵𝐶 to 2 decimal places.

Q12:

Find the area of the green part of the diagram, given that 𝑚∠𝐶𝐴𝐵=77∘, 𝑚∠𝐵𝐶𝐴=57∘, and 𝐶𝐵=19cm. Give the answer to the nearest square centimeter.

Q13:

𝐴𝐵𝐶 is an equilateral triangle inscribed in circle 𝑀 with a radius of 95 cm. Find the area of each shaded segment giving the answer to the nearest square centimeter.

Q14:

𝐴𝐵𝐶 is a triangle where the ratio between 𝑚∠𝐴,
𝑚∠𝐵, and 𝑚∠𝐶 is 5∶4∶3 and the radius of the circumcircle is
6 cm. Find the area of 𝐴𝐵𝐶 giving
the answer to two decimal places.

Q15:

Two planes leave the same airport at the same time. One flies on a bearing of
20∘ at 500 miles per hour. The other flies on a bearing
of 30∘ at 600 miles per hour. How far apart are the planes
after 2 hours?

Q16:

The shown diagram represents a satellite calculating the distances and the angle.
Find the distance between the two cities. Give your answer to one decimal place. Note that the diagram is not to scale.

Q17:

A 113-foot tower is located on a hill that is inclined
34∘ to the horizontal,
as shown in the figure.
A guy wire is to be attached to the top of the tower and anchored at a point
98 feet
uphill from the base of the tower.
Find the length of wire needed.

Q18:

A man ran 24 km
down a straight road. He then turned 139∘ and ran a further 9 km along another straight road. Find, to the nearest kilometer, the shortest distance between his start and end points.

Q19:

An airplane flies 220 miles
on a bearing of 40∘
and then changes course and flies 180 miles
on a bearing of 170∘.
How far is the plane from its starting point and on what bearing? Give your answers to one decimal place.

A172.9 miles, 52.9∘

B172.9 miles, 92.9∘

C52.9 miles, 92.9∘

D362.9 miles, 102.5∘

E6,141.9 miles, 37.1∘

Q20:

Two boats leave a harbor in different directions. One travels due west at
30 mph,
and the other travels on a bearing of
295∘ at
42 mph. How far apart are they after
90 minutes?

Q21:

Two ships left a port at the same time. One traveled on a bearing of 320∘ at
18 mph, while the other traveled on a bearing of
194∘ at 22 mph.
Find the distance between the ships after 10 hours.
Give your answer to the nearest mile.

Q22:

A plane follows a straight path at a steady speed of 600 miles per hour.
After 1 hour30 minutes, the pilot makes a course
correction of 10∘,
and the flight continues at the same speed, following a straight path, for another 2 hours. How far is the plane from its
initial position? Give your answer to the nearest mile.

Q23:

Philadelphia is 140 miles from Washington, DC. Washington, DC, is 442 miles from Boston. Boston is 315 miles from Philadelphia. Find the angles in the triangle whose vertices are at Philadelphia, Washington, DC, and Boston.

A9.06∘, 20.74∘, 150.20∘

B2.17∘, 62.12∘, 115.71∘

C3.88∘, 115.71∘, 60.41∘

D62.12∘, 60.41∘, 57.47∘

E62.12∘, 20.74∘, 97.14∘

Q24:

Los Angeles is 1,744 miles from Chicago, Chicago is 712 miles from New York, and New York is 2,451 miles from Los Angeles. Find the angles in the triangle with its vertices at the three cities.

A4.66∘, 5.73∘, 169.61∘

B2.33∘, 19.44∘, 158.23∘

C84.27∘, 2.33∘, 93.4∘

D2.33∘, 5.73∘, 171.94∘

E2.33∘, 11.64∘, 166.03∘

Q25:

Two sides of a triangular flower bed measure 40 feet and 65 feet with an angle of
50∘ between
them. Calculate the length of the third side, giving your answer to one decimal place.